Write down the co-factors of the elements of the first row of the following determinant and hence evaluate the determinant
`|{:(1,3,-3),(2,-1,0),(4,-2,5):}|`

Answer

To solve the problem, we need to find the cofactors of the elements of the first row of the determinant and then evaluate the determinant itself.

Given determinant:
\[ D = \begin{vmatrix} 1 & 3 & -3 \\ 2 & -1 & 0 \\ 4 & -2 & 5 \end{vmatrix} \]

### Step 1: Calculate the Cofactor \( A_{11} \)

The cofactor \( A_{11} \) is calculated as follows:

1. **Minor of \( A_{11} \)**: Remove the first row and first column:
\[ \text{Minor} = \begin{vmatrix} -1 & 0 \\ -2 & 5 \end{vmatrix} = (-1) \cdot 5 - (0) \cdot (-2) = -5 \]

2. **Cofactor**:
\[ A_{11} = (-1)^{1+1} \cdot \text{Minor} = 1 \cdot (-5) = -5 \]

### Step 2: Calculate the Cofactor \( A_{12} \)

The cofactor \( A_{12} \) is calculated as follows:

1. **Minor of \( A_{12} \)**: Remove the first row and second column:
\[ \text{Minor} = \begin{vmatrix} 2 & 0 \\ 4 & 5 \end{vmatrix} = (2) \cdot (5) - (0) \cdot (4) = 10 \]

2. **Cofactor**:
\[ A_{12} = (-1)^{1+2} \cdot \text{Minor} = -1 \cdot 10 = -10 \]

### Step 3: Calculate the Cofactor \( A_{13} \)

The cofactor \( A_{13} \) is calculated as follows:

1. **Minor of \( A_{13} \)**: Remove the first row and third column:
\[ \text{Minor} = \begin{vmatrix} 2 & -1 \\ 4 & -2 \end{vmatrix} = (2)(-2) - (-1)(4) = -4 + 4 = 0 \]

2. **Cofactor**:
\[ A_{13} = (-1)^{1+3} \cdot \text{Minor} = 1 \cdot 0 = 0 \]

### Step 4: Write the Cofactors

The cofactors of the first row are:
- \( A_{11} = -5 \)
- \( A_{12} = -10 \)
- \( A_{13} = 0 \)

### Step 5: Evaluate the Determinant

Using the cofactors, we can evaluate the determinant:
\[ D = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} \]
Substituting the values:
\[ D = 1 \cdot (-5) + 3 \cdot (-10) + (-3) \cdot 0 \]
\[ D = -5 - 30 + 0 = -35 \]

### Final Result

The value of the determinant is:
\[ D = -35 \]